The slides of my introductory course to relative toposes recently given at the Lagrange Center are now available:
Next week, on the 8th and 9th of September, the Ninth Symposium on Compositional Structures (SYCO 9) is taking place in Como.
I have been invited to give a talk on this occasion. The title of my presentation is Relative toposes as a generalization of locales. The abstract is as follows:
The aim of this talk is to present a way for representing relative toposes which naturally generalizes the construction of the topos of sheaves on a locale, and which is particularly effective for describing the morphisms between relative toposes in a concrete way. Our theoretical framework is based on the language of stacks and fibred sites, and provides, amongst other things, a unified setting for investigating the relationships between Grothendieck toposes as built from sites and elementary toposes as built from triposes.
Looking forward to seeing many of you in Como!
I am glad to announce the first version of our joint work with Riccardo Zanfa on relative toposes:
Here is the abstract:
We introduce new foundations for relative topos theory based on stacks. One of the central results in our theory is an adjunction between the category of toposes over the topos of sheaves on a given site (C,J) and that of C-indexed categories. This represents a wide generalization of the classical adjunction between presheaves on a topological space and bundles over it, and allows one to interpret several constructions on sheaves and stacks in a geometrical way; in particular, it leads to fibrational descriptions of direct and inverse images of sheaves and stacks, as well as to a geometric understanding of the sheafification process. It also naturally allows one to regard any Grothendieck topos as a ‘petit’ topos associated with a ‘gros’ topos, thereby providing an answer to a problem posed by Grothendieck in the seventies. Another key ingredient in our theory is a notion of relative site, which allows one to represent arbitrary geometric morphisms towards a fixed base topos of sheaves on a site as structure morphisms induced by relative sites over that site.
We shall progressively release expanded versions of this text contaning new developments in the directions sketched in the introduction.
This work has been recently presented at the conference Toposes online:
The following video focuses on the specialization of the fundamental adjunction in the setting of presheaves (or discrete fibrations):
The video of my yesteday talk on relative toposes is already available from YouTube:
Thanks again to the organizers for their invitation, and for making the recording available so quickly!
Next Tuesday I shall give a talk for ItaCa Fest 2021 on my work in progress with Riccardo Zanfa providing new foundations for relative topos theory based on stacks:
Here is the Zoom link to attend the talk: https://zoom.us/j/94880770089?pwd=clgxK2VkVEE5Ymw5ME1QWktiWExUZz09
Thanks to the ItaCa Fest organizers for this invitation, and looking forward to seeing many of you there!